A337655 -id:A337655 - cp's OEIS frontend

A337659 Triangular array read by rows: T(n,k) = A337655(n) + A337655(k), for 1 <= k <= n.

2, 3, 4, 6, 7, 10, 8, 9, 12, 14, 16, 17, 20, 22, 30, 23, 24, 27, 29, 37, 44, 32, 33, 36, 38, 46, 53, 62, 51, 52, 55, 57, 65, 72, 81, 100, 69, 70, 73, 75, 83, 90, 99, 118, 136, 91, 92, 95, 97, 105, 112, 121, 140, 158, 180, 102, 103, 106, 108, 116, 123, 132, 151, 169, 191, 202

Offset: 1

N. J. A. Sloane, Oct 03 2020

This is the lower triangular part of the addition table from A337655, read by rows.

Sequences A337659, A337660, A337661, A337662 arise from the addition and multiplication tables in A337655, each one described in two ways. Perhaps someone could help by creating the analogous sets of four sequences for the addition and multiplication tables in the closely related sequences A337656 and A337946.

			The addition table from A337655 begins:
   2,  3,  6,  8,  16,  23,  32,  51,  69,  91, ...
   3,  4,  7,  9,  17,  24,  33,  52,  70,  92, ...
   6,  7, 10, 12,  20,  27,  36,  55,  73,  95, ...
   8,  9, 12, 14,  22,  29,  38,  57,  75,  97, ...
  16, 17, 20, 22,  30,  37,  46,  65,  83, 105, ...
  23, 24, 27, 29,  37,  44,  53,  72,  90, 112, ...
  32, 33, 36, 38,  46,  53,  62,  81,  99, 121, ...
  51, 52, 55, 57,  65,  72,  81, 100, 118, 140, ...
  69, 70, 73, 75,  83,  90,  99, 118, 136, 158, ...
  91, 92, 95, 97, 105, 112, 121, 140, 158, 180, ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A337655, A337660, A337661, A337662.

A337661 Triangular array read by rows: T(n,k) = A337655(n) * A337655(k), for 1 <= k <= n.

Original entry on oeis.org

1, 2, 4, 5, 10, 25, 7, 14, 35, 49, 15, 30, 75, 105, 225, 22, 44, 110, 154, 330, 484, 31, 62, 155, 217, 465, 682, 961, 50, 100, 250, 350, 750, 1100, 1550, 2500, 68, 136, 340, 476, 1020, 1496, 2108, 3400, 4624, 90, 180, 450, 630, 1350, 1980, 2790, 4500, 6120, 8100

Offset: 1

N. J. A. Sloane, Oct 03 2020

This is the lower triangular part of the multiplication table from A337655, read by rows.

			The multiplication table from A337655 begins:
   1,   2,   5,   7,   15,   22,   31,   50,   68,   90, ...
   2,   4,  10,  14,   30,   44,   62,  100,  136,  180, ...
   5,  10,  25,  35,   75,  110,  155,  250,  340,  450, ...
   7,  14,  35,  49,  105,  154,  217,  350,  476,  630, ...
  15,  30,  75, 105,  225,  330,  465,  750, 1020, 1350, ...
  22,  44, 110, 154,  330,  484,  682, 1100, 1496, 1980, ...
  31,  62, 155, 217,  465,  682,  961, 1550, 2108, 2790, ...
  50, 100, 250, 350,  750, 1100, 1550, 2500, 3400, 4500, ...
  68, 136, 340, 476, 1020, 1496, 2108, 3400, 4624, 6120, ...
  90, 180, 450, 630, 1350, 1980, 2790, 4500, 6120, 8100, ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A337655, A337659, A337660, A337662.

A337662 Multiplication table from A337655, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 5, 4, 5, 7, 10, 10, 7, 15, 14, 25, 14, 15, 22, 30, 35, 35, 30, 22, 31, 44, 75, 49, 75, 44, 31, 50, 62, 110, 105, 105, 110, 62, 50, 68, 100, 155, 154, 225, 154, 155, 100, 68, 90, 136, 250, 217, 330, 330, 217, 250, 136, 90, 101, 180, 340, 350, 465, 484, 465, 350, 340, 180

Offset: 1

N. J. A. Sloane, Oct 04 2020

			The multiplication table from A337655 begins:
   1,   2,   5,   7,   15,   22,   31,   50,   68,   90, ...
   2,   4,  10,  14,   30,   44,   62,  100,  136,  180, ...
   5,  10,  25,  35,   75,  110,  155,  250,  340,  450, ...
   7,  14,  35,  49,  105,  154,  217,  350,  476,  630, ...
  15,  30,  75, 105,  225,  330,  465,  750, 1020, 1350, ...
  22,  44, 110, 154,  330,  484,  682, 1100, 1496, 1980, ...
  31,  62, 155, 217,  465,  682,  961, 1550, 2108, 2790, ...
  50, 100, 250, 350,  750, 1100, 1550, 2500, 3400, 4500, ...
  68, 136, 340, 476, 1020, 1496, 2108, 3400, 4624, 6120, ...
  90, 180, 450, 630, 1350, 1980, 2790, 4500, 6120, 8100, ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A337655, A337659, A337661, A337662.

A337660 Addition table from A337655, read by antidiagonals.

Original entry on oeis.org

2, 3, 3, 6, 4, 6, 8, 7, 7, 8, 16, 9, 10, 9, 16, 23, 17, 12, 12, 17, 23, 32, 24, 20, 14, 20, 24, 32, 51, 33, 27, 22, 22, 27, 33, 51, 69, 52, 36, 29, 30, 29, 36, 52, 69, 91, 70, 55, 38, 37, 37, 38, 55, 70, 91, 102, 92, 73, 57, 46, 44, 46, 57, 73, 92, 102, 125, 103, 95, 75, 65, 53, 53, 65, 75, 95, 103, 125

Offset: 1

N. J. A. Sloane, Oct 04 2020

			The addition table from A337655 begins:
   2,  3,  6,  8,  16,  23,  32,  51,  69,  91, ...
   3,  4,  7,  9,  17,  24,  33,  52,  70,  92, ...
   6,  7, 10, 12,  20,  27,  36,  55,  73,  95, ...
   8,  9, 12, 14,  22,  29,  38,  57,  75,  97, ...
  16, 17, 20, 22,  30,  37,  46,  65,  83, 105, ...
  23, 24, 27, 29,  37,  44,  53,  72,  90, 112, ...
  32, 33, 36, 38,  46,  53,  62,  81,  99, 121, ...
  51, 52, 55, 57,  65,  72,  81, 100, 118, 140, ...
  69, 70, 73, 75,  83,  90,  99, 118, 136, 158, ...
  91, 92, 95, 97, 105, 112, 121, 140, 158, 180, ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A337655, A337659, A337661, A337662.

A337658 Let M_k denote the addition table for the first k terms of A337655. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

Original entry on oeis.org

1, 5, 11, 13, 15, 18, 19, 21, 25, 26, 28, 31, 34, 35, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 54, 56, 58, 59, 60, 61, 63, 64, 66, 67, 68, 71, 74, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 93, 94, 96, 98, 101, 104, 107, 109, 110, 111, 113, 114, 115, 117, 119, 120, 122, 124, 127

Offset: 1

Jean-Paul Delahaye, Oct 01 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A337658

Cf. A337655, A337656, A337657.

A337656 a(1)=0; thereafter, a(n) is the smallest number such that the addition table for (a(1),...,a(n)) contains n(n+1)/2 different entries, and the multiplication table for (a(2),...,a(n)) contains n(n-1)/2 different nonzero entries (the maximum possible in both cases).

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 30, 44, 65, 91, 107, 122, 147, 196, 230, 263, 329, 375, 397, 472, 545, 596, 677, 716, 770, 931, 1007, 1059, 1201, 1330, 1415, 1484, 1570, 1688, 1934, 1982, 2229, 2391, 2550, 2646, 2837, 3078, 3138, 3208, 3383, 3560, 3596, 3930, 4257, 4421, 4673

Offset: 1

N. J. A. Sloane, Oct 01 2020

nonn

Inspired by A337655.

The distinct entries in the addition table for a(1),...,a(n), when sorted into increasing order, converge to a sequence which consists of all positive numbers except 5, 9, 11, 16, 17, 18, 22, 25, 26, 28, ... (see A337657).

Rémy Sigrist, Table of n, a(n) for n = 1..3000 (first 87 terms from N. J. A. Sloane)
Rémy Sigrist, C++ program for A337656

Cf. A337655, A337657, A337658.

Corrected and extended by Jean-Paul Delahaye, Oct 01 2020

A337657 Let M_k denote the addition table for the first k terms of A337656. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

Original entry on oeis.org

5, 9, 11, 16, 17, 18, 22, 25, 26, 28, 29, 34, 35, 36, 38, 39, 41, 43, 46, 48, 49, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 67, 69, 70, 71, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 90, 93, 96, 97, 99, 100, 101, 102, 104, 105, 106, 112, 113, 115, 116, 117, 118, 120, 124, 126, 128

Offset: 1

N. J. A. Sloane, Oct 01 2020

nonn

Note that if A337656(k+1) = t, all entries in M_{k+1} that are not entries in M_k are >= t.

			The addition table, M_9:
   + |  0  1  3  7 12 20 30  44  65
-----+-----------------------------
   0 |  0
   1 |  1  2
   3 |  3  4  6
   7 |  7  8 10 14
  12 | 12 13 15 19 24
  20 | 20 21 23 27 32 40
  30 | 30 31 33 37 42 50 60
  44 | 44 45 47 51 56 64 74  88
  65 | 65 66 68 72 77 85 95 109 130

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A337657

Cf. A337655, A337656, A337658.

A337946 a(1)=1; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n*(n+1) different entries (the maximum possible).

Original entry on oeis.org

1, 3, 7, 12, 22, 30, 47, 61, 85, 113, 126, 177, 193, 246, 279, 321, 341, 428, 499, 571, 616, 686, 754, 854, 975, 1052, 1150, 1317, 1376, 1457, 1513, 1664, 1761, 1961, 2307, 2434, 2591, 2795, 2843, 3057, 3226, 3405, 3508, 3776, 3930, 4023, 4196, 4575, 4731

Offset: 1

Peter Kagey, Oct 02 2020

nonn

			The addition table of a(k) for k=1..5:
   + | 1 3  7 12 22
  ---+-------------
   1 | 2 4  8 13 23
   3 |   6 10 15 25
   7 |     14 19 29
  12 |        24 34
  22 |           44
The multiplication table of a(k) for k=1..5:
   * | 1 3  7  12  22
  ---+---------------
   1 | 1 3  7  12  22
   3 |   9 21  36  66
   7 |     49  84 154
  12 |        144 264
  22 |            484
These two tables contain the 5*(5+1) = 30 values {1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 19, 21, 22, 23, 24, 25, 29, 34, 36, 44, 49, 66, 84, 144, 154, 264, 484}.

Peter Kagey, Table of n, a(n) for n = 1..1000
Peter Kagey, Haskell program

Cf. A005282 (addition table), A066720 (multiplication table), A337655, A337656, A337947.

j={k=1};Do[While[l=Join[j,{++k}];g=Union[Sort/@Tuples[l,{2}]];p=Times@@#&/@g;s=Total/@g;!SameQ@@Flatten[{Length@Union@Flatten@{p,s},Length@l(Length@l+1)}]];j=Join[j,{k}];k=Last@j,48];j (* Giorgos Kalogeropoulos, Nov 16 2021 *)

A338012 a(1)=0; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n^2 different entries (the maximum possible).

Original entry on oeis.org

0, 3, 4, 10, 18, 23, 34, 55, 67, 93, 95, 120, 149, 166, 228, 271, 351, 398, 439, 505, 563, 611, 732, 771, 806, 924, 1052, 1121, 1278, 1412, 1586, 1654, 1875, 2012, 2245, 2341, 2445, 2616, 2819, 2920, 3034, 3322, 3518, 3754, 3918, 4016, 4311, 4649, 4848, 5321

Offset: 1

Peter Kagey, Oct 06 2020

nonn

			Addition table begins:
   + |  0  3  4 10 18 23  34  55  67
-----+-------------------------------
   0 |  0
   3 |  3  6
   4 |  4  7  8
  10 | 10 13 14 20
  18 | 18 21 22 28 36
  23 | 23 26 27 33 41 46
  34 | 34 37 38 44 52 57  68
  55 | 55 58 59 65 73 78  89 110
  67 | 67 70 71 77 85 90 101 122 134
Multiplication table begins:
   * | 0   3   4  10   18   23   34   55   67
-----+---------------------------------------
   0 | 0
   3 | 0   9
   4 | 0  12  16
  10 | 0  30  40 100
  18 | 0  54  72 180  324
  23 | 0  69  92 230  414  529
  34 | 0 102 136 340  612  782 1156
  55 | 0 165 220 550  990 1265 1870 3025
  67 | 0 201 268 670 1206 1541 2278 3685 4489

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A337655, A337656, A337946.

Cf. A338013, A338014, A338015, A338016, A338017.

cp's OEIS Frontend

A337659 Triangular array read by rows: T(n,k) = A337655(n) + A337655(k), for 1 <= k <= n.

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A337661 Triangular array read by rows: T(n,k) = A337655(n) * A337655(k), for 1 <= k <= n.

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A337662 Multiplication table from A337655, read by antidiagonals.

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A337660 Addition table from A337655, read by antidiagonals.

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A337658 Let M_k denote the addition table for the first k terms of A337655. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

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A337656 a(1)=0; thereafter, a(n) is the smallest number such that the addition table for (a(1),...,a(n)) contains n*(n+1)/2 different entries, and the multiplication table for (a(2),...,a(n)) contains n*(n-1)/2 different nonzero entries (the maximum possible in both cases).

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Extensions

A337657 Let M_k denote the addition table for the first k terms of A337656. M_k contains exactly k*(k+1)/2 distinct numbers, and these numbers are a subset of the entries in M_{k+1}. The present sequence consists of the numbers that never appear in any M_k.

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A337946 a(1)=1; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n*(n+1) different entries (the maximum possible).

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Mathematica

A338012 a(1)=0; thereafter, a(n) is the smallest number such that the addition and multiplication tables for (a(1),...,a(n)) together contain n^2 different entries (the maximum possible).

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A337656 a(1)=0; thereafter, a(n) is the smallest number such that the addition table for (a(1),...,a(n)) contains n(n+1)/2 different entries, and the multiplication table for (a(2),...,a(n)) contains n(n-1)/2 different nonzero entries (the maximum possible in both cases).